Convergence of Functionals of Sums of R.V.s to Local Times of Fractional Stable Motions

• Published in: The Annals of probability 2004-07, Vol.32 (3), p.1771-1795
• Main Author: Jeganathan, P.
• Format: Article
• Language: English
• Subjects: 60F05; 60G18; 60J55; 62J02; 62M10; Absolute value; Approximation; Brownian motion; Distribution theory; Eigenfunctions; Exact sciences and technology; Fourier transformations; fractional ARIMA process; fractional Brownian motion; Fractional stable motion; functionals of sums of fractional ARIMA; General topics; heavy tailed distributions; Inference from stochastic processes; time series analysis; Integers; local time; Markov processes; Mathematical analysis; Mathematical constants; Mathematical models; Mathematical theorems; Mathematics; Perceptron convergence procedure; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Statistics; Studies; Variables; weak convergence to local times
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/009117904000000658

Consider a sequence Xk=∑j=0 ∞cjξk-j,k≥ 1, where cj,j≥ 0, is a sequence of constants and ξj, -∞ < j < ∞, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0 < α ≤ 2. Let Sk=∑j=1 kXj. Under suitable conditions on the constants cjit is known that for a suitable normalizing constant γn, the partial sum process γn -1S[nt]converges in distribution to a linear fractional stable motion (indexed by α and H, 0 < H < 1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process Xk. In this paper it is established that the process n-1βn∑k=1 [nt]f(βn(γn -1Sk+x)) converges in distribution to (∫-∞ ∞f(y)dy)L(t, -x), where L(t, x) is the local time of the linear fractional stable motion, for a wide class of functions f(y) that includes the indicator functions of bounded intervals of the real line. Here βn→ ∞ such that n-1βn→ 0. The only further condition that is assumed on the distribution of ξ1is that either it satisfies the Cramér's condition or has a nonzero absolutely continuous component. The results have motivation in large sample inference for certain nonlinear time series models.